Betti Numbers of Finite Volume Orbifolds 10

BETTI NUMBERS OF FINITE VOLUME ORBIFOLDS IDDO SAMET Abstract. We prove that the Betti numbers of a negatively curved orbifold are linearly bounded by its volume, generalizing a theorem of Gromov which establishes this bound for manifolds. An immediate corollary is that Betti numbers of a lattice in a rank-one Lie group are linearly bounded by its co-volume. 1. Introduction Let X be a Hadamard manifold, i.e. a connected, simply-connected, complete Riemannian manifold of non-positive curvature normalized such that −1 ≤ K ≤ 0. Let Γ be a discrete subgroup of Isom(X). If Γ is torsion-free then X/Γ is a Riemannian manifold. If Γ has torsion elements, then X/Γ has a structure of an orbifold. In both cases, the Riemannian structure defines the volume of X/Γ. An important special case is that of symmetric spaces X = K\G, where G is a connected semisimple Lie group without center and with no compact factors, and K is a maximal compact subgroup. Then X is non-positively curved, and G is the connected component of Isom(X). In this case, X/Γ has finite volume iff Γ is a lattice in G. The curvature of X is non-positive if rank(G) ≥ 2, and is negative if rank(G) = 1. In many cases, the volume of negatively curved manifold controls the complexity of its topology. A manifestation of this phenomenon is the celebrated theorem of Gromov [3, 10] stating that if X has sectional curvature −1 ≤ K < 0 then n∑<ul style="display: flex;"><li style="flex:1">(1) </li><li style="flex:1">bi(X/Γ) ≤ Cn · vol(X/Γ), </li></ul>i=0 where bi are Betti numbers w.r.t. to coefficients in any field, and Cn is a constant depending only on n = dim(X). This inequality also holds when −1 ≤ K ≤ 0 if X is analytic and has no Euclidean factors. Date: November 2012. The research was done as part of the author’s PhD at the Hebrew University under the supervision of Prof. Gelander and was supported by the European Research Council (ERC) grant of Prof. Gelander / grant agreement 203418. 1<ul style="display: flex;"><li style="flex:1">BETTI NUMBERS OF FINITE VOLUME ORBIFOLDS </li><li style="flex:1">2</li></ul>It should be noted, that since X is a K(Γ, 1), the bi’s are the Betti number of the homology of the group Γ. In certain cases, volume imposes much stronger restriction on topology. Namely, it was proved by Gelander [8] that if X is a symmetric space of non-compact type and Γ is a non-uniform torsion-free arithmetic lattice then X/Γ is homotopy-equivalent to a simplicial complex whose complexity is restricted: the number of vertices in this complex is bounded by a constant (depending on X) times the volume of X/Γ, and the valence of each vertex is bounded by a constant depending only on X. This equivalence immediately implies Gromov’s theorem. By Margulis’s arithmeticity theorem, if rank(X) ≥ 2 then every lattice Γ is arithmetic, and thus Gelander’s theorem holds for all such spaces. Gelander has also proved that if Γ torsion-free, then Γ has a presentation for which the number of generators and the total length of the relations is bounded linearly by the volume of X/Γ (with constants depending on X). Recently, Gelander [9] extended this result and proved that even for lattices with torsion, the minimal number of generators of Γ is bounded from above by a constant (depending on G) times the volume of X/Γ. In particular, this provides a linear bound to the first Betti number of Γ. The main theorem proved in this paper is a generalization of Gromov’s theorem to the case where Γ has torsion, in other words, to orbifolds of the form X/Γ. We prove Theorem 1.1. Let X be a Hadamard manifold of dimension n with sectional curvature −1 ≤ K < 0. Then for every discrete group Γ < Isom(X), ∑<ul style="display: flex;"><li style="flex:1">(2) </li><li style="flex:1">rk Hi(X/Γ, A) ≤ Cn · vol(X/Γ), </li></ul>iwhere the coefficient ring A is an integral domain of characteristic 0, and Cn is a constant depending only on n. There are several (co)-homology theories for orbifolds which aim at capturing the homology of the space, as well as that of the group Γ. Our technique is mainly geometric; we therefore treat X/Γ as a topological quotient space, and the homology is that of this space. Furthermore, the action of finite subgroups of Γ introduces torsion to homology groups, which may depend on the structure of these finite groups. Since we are not able to bound the size of finite groups by the orbifold volume, our result is restricted to homology with coefficients in an integral domain with characteristic 0. <ul style="display: flex;"><li style="flex:1">BETTI NUMBERS OF FINITE VOLUME ORBIFOLDS </li><li style="flex:1">3</li></ul>The case A = Q is of particular importance. The homology of the stabilizers in G of points of X is trivial, because stabilizers are finite groups. It follows that Hi(X/Γ, Q) = Hi(Γ, Q) (see e.g. [6], p. 174). Hence, we can restate (2) as a bound on Betti numbers (with rational coefficients) of the group Γ: ∑bi(Γ, Q) ≤ Cn · vol(X/Γ). iTheorem 1.1 and Gromov’s theorem can be rephrased as an asymptotical statement: if Γn is a sequence of groups of isometries acting discretely on X (with the appropriate assumptions) then the growth of bi(Γn) is at most linear in vol(X/Γn). A theorem of Lu¨ck [12] describes the asymptotic behavior of a nested sequence (Γn)n≥1 of normal finite index subgroups of Γ, assuming that ∩X/Γ is compact and Γn = 1. In this case, bi(X/Γn) <ul style="display: flex;"><li style="flex:1">(3) </li><li style="flex:1">lim </li><li style="flex:1">= b(i2)(X; Γ), </li></ul>n→∞ vol(X/Γn) where the right-hand-side is the L2-Betti number. Hence, the growth is linear exactly when b(i2)(X; Γ) is non-zero. It was proved by Olbrich [15] that if X is a symmetric space, then b(i2) is zero unless, perhaps, i = dim(X)/2. In a recent joint work with Abert, Bergeron, Biringer, Gelander, Nikolov and Raimbault [1,2] we prove (3) holds for any sequence of uniform irreducible lattices (Γn)n≥1 (possibly with torsion) in a semisimple Lie group G, assuming that vol(X/Γn) → ∞ and that G has property (T) and rank ≥ 2. However, one cannot expect (3) to hold for rankone locally symmetric spaces (whether manifolds or orbifolds). For ninstance, if Γ is a uniform lattice in Isom(H ) that surjects on the free group of rank 2, then finite index subgroups of Γ corresponding to subgroups of Z∗Z have first Betti numbers that grow linearly with volume, whereas b(12) = 0. Characteristic p. It is natural to whether Theorem 1.1 holds for coefficients with positive characteristic, and particularly for coefficients in the field Fp. The author does not know of any counterexample in this case. In fact, there is but a single place in the proof that relies on the characteristic assumption, namely Proposition 4.11. Proving this proposition for coefficients in Fp boils down to the problem of uniformly bounding the Fp-Betti numbers of a spherical orbifold, as posed in the following question: <ul style="display: flex;"><li style="flex:1">BETTI NUMBERS OF FINITE VOLUME ORBIFOLDS </li><li style="flex:1">4</li></ul>Does there exist a constant D depending on k (and possibly on p?) such that for every finite group G acting linearly on a sphere Sk, and for every i, bi(Sk/G, Fp) ≤ D? More precisely, an affirmative answer to the question for spheres of dimensions ≤ n − 1 would imply that Theorem 1.1 holds with coefficients in Fp for manifolds of dimension n. In particular, we can obtain the following partial results: (1) The theorem holds with Fp coefficients for manifolds of dimensions 2 or 3 (with a constant Cn independent of p). This is because — up to homeomorphism — there are only finitely many possible quotients of 1-spheres and 2-spheres by finite groups1. (2) If p does not divide the order of any finite subgroup of Γ, then the inequality (2) holds for A = Fp (with the constant Cn of the characteristic 0 case). This follows from the fact that bi(Sk/G, Fp) ≤ 1 whenever p is prime to the order of G (see remark at the end of section 4). Acknowledgments. It is my pleasure to thank Tsachik Gelander, my Ph.D. advisor, for his support, encouragement and advice. I also thank the referee for carefully reading the manuscript, and for his invaluable suggestions for improving the exposition. 2. Preliminaries 2.1. Notation. Let X be an n-dimensional Hadamard manifold, i.e. a connected, simply-connected complete Riemannian manifold of nonpositive sectional curvature. We assume that the curvature is bounded and that the metric is normalized such that −1 ≤ K ≤ 0. If K < 0 we say the manifold is negatively curved. Most of our results apply to this case. 2.1.1. We introduce notation and review some facts about isometries of Hadamard spaces. A standard reference for this is [5, Ch. II.6] or [3, §6]. Let γ be an isometry of X. The displacement function dγ : X → R≥0, dγ(x) = d(γx, x), is a convex function in the sense that for every geodesic c : R → X, dγ ◦ c is a convex function. The set of minimal displacement is defined Min(γ) = {x ∈ X : dγ(x) = inf dγ}; 1The underlying space of a 1-dimensional spherical orbifold is 1-sphere or an interval, and the underlying space of a 2-dimensional spherical orbifold is a 2-sphere, a disc, or a projective plane. <ul style="display: flex;"><li style="flex:1">BETTI NUMBERS OF FINITE VOLUME ORBIFOLDS </li><li style="flex:1">5</li></ul>it is a closed and convex subset of X. An isometry of X is elliptic if it fixes a point in X, hyperbolic if it does not fix a point in X but dγ attains minimum in X, and parabolic otherwise. The first two types are called semisimple. An isometry is elliptic (resp. hyperbolic, parabolic) iff any a positive power of it is elliptic (resp. hyperbolic, parabolic). If γ is a non-trivial elliptic isometry then Min(γ) is a complete totally geodesic submanifold of lower dimension in X. If γ is hyperbolic and K < 0 then Min(γ) is a geodesic — the axis of γ — on which γ acts by translations. If γ is a parabolic isometry then it fixes a point in the boundary of points at infinity. Explicitly, there is a geodesic ray +c : R → X such that c and γ c are asymptotic, i.e., d(c(t), γ c(t)) is uniformly bounded. If K < 0 then γ fixes a unique point at infinity. Let C be a closed convex subset of X, and let x ∈ X. Since X has non-positive curvature, there is a unique point πC(x) ∈ C that is closest to x in C. This is called the projection to C. The projection is distance decreasing, i.e. d(πC(x), πC(y)) ≤ d(x, y). Now suppose C is γ-invariant. Then γπC(x) = πC(γx). It follows that dγ(πC(x)) ≤ dγ(x). In particular, Min(γ) ∩ C is non-empty. <ul style="display: flex;"><li style="flex:1">If γ0, γ1 are commuting semisimple isometries, then γi keeps Min(γ1−i </li><li style="flex:1">)</li></ul>invariant (i = 0, 1). It follows that Min(γ0) ∩ Min(γ1) is non-empty, and kept invariant by both isometries. More generally, if A is a set of ∩commuting semisimple isometries then γ∈A Min(γ) is non-empty and A-invariant. In particular, if K < 0 then two commuting isometries have the same axis. 2.1.2. Let Γ be a group of isometries of X acting properly discontinuously, that is, a subgroup of Isom(X) which is discrete with respect to the compact-open topology. If Γ is torsion-free, then X/Γ has a structure of Riemannian manifold. More generally, Γ may have finite point-stabilizers, and X/Γ is endowed a structure of a Riemannian orbifold. An orbifold has an atlas of maps locally identifying it with a quotient of an open set in X by a finite group of isometries. Orbifolds were originally introduced and studied by Satake [17] who named them V-manifolds. 2.1.3. We have the canonical projection π : X → X/Γ. Let inj-rad(x), x ∈ X/Γ, be the local injectivity radius at x. Let us denote dΓ(x) = 12infγ∈Γ\{1} dγ(x). We recall that inj-rad(x) = dΓ(x˜), with π(x˜) = x. The ε-thick part of X/Γ is defined ε(X/Γ) = {x ∈ X/Γ : inj-rad(x) ≥ }. ≥ε 2<ul style="display: flex;"><li style="flex:1">BETTI NUMBERS OF FINITE VOLUME ORBIFOLDS </li><li style="flex:1">6</li></ul>It is the image of the set {x ∈ X : dΓ(x) ≥ ε} under the canonical projection. We remark that if X is a point, its injectivity radius is infinity. 2.1.4. We will make all volume calculations using discrete estimates. Say a set N ⊂ X is δ-discrete if d(x, y) ≥ δ for x = y ∈ N. For ε > 0 ̸we define ε − ess-vol(X/Γ) as the supremal cardinality of a 2ε-discrete set of points N in the 2ε-thick part of X/Γ. This is the supremum of the number of disjoint injected balls of radius ε. It is reasonable to set ε − ess-vol(point) = 1; this is consistent with definitions for positive dimension. This definition is introduced in [3, §10.4]. Clearly, ε − ess-vol(X/Γ) is non-decreasing as ε decreases. In the case dim X > 0, it is easy to verify the useful inequality εk · ε − ess-vol(X/Γ) ≤ − ess-vol(X/Γ). kSince curvature is non-positive, a ball of radius ε in X has volume ≥ V (ε, n) — the volume of a ball of radius ε in Euclidean n-space. Thus we have ε − ess-vol(X/Γ) ≤ V (ε, n)−1 · vol(X/Γ). We will utilize these two inequalities freely without further reference. 2.1.5. For a point x ∈ X, let Γx be the stabilizer of x in Γ. A subgroup of Γ is finite if and only if it is contained in Γx, for some x ∈ X. For a submanifold Y ⊆ X, we denote ΓY = {γ ∈ Γ : γY = Y }, and Γ1Y = {γ ∈ ΓY : γ|Y = 1}. The latter is a finite group. If Γ1Y is not trivial, then ΓY cannot be identified as a subgroup of Isom(Y ). However, the action of ΓY on Y factors through ΓY /Γ1Y . In this case, the terms Y/ΓY and dΓ , and consequentially inj-rad and Yε − ess-vol(Y/ΓY ), will refer to — by abuse of notation — the action ΓY /Γ1Y on Y . 2.2. Margulis lemma. We denote ∆ε(x) = {γ ∈ Γ : dγ(x) < ε} and Γε(x) = ⟨∆ε(x)⟩. The following statement about groups generated by “small motions” is an amalgamation of the Margulis Lemma [3] and the Jordan Theorem [11]. Theorem 2.1. Let X be a Hadamard manifold of dimension n with −1 ≤ K ≤ 0. There are constants depending only on n, εn > 0 and mn ∈ N such that for every discrete subgroup Γ < Isom(X) and every x ∈ X, Γε (x) contains a normal nilpotent subgroup N of index ≤ mn. nIf Γε (x) is finite, N is in fact abelian. n<ul style="display: flex;"><li style="flex:1">BETTI NUMBERS OF FINITE VOLUME ORBIFOLDS </li><li style="flex:1">7</li></ul>We henceforth refer to the nilpotent subgroup stipulated by this theorem as “the normal nilpotent subgroup of Γε(x)”. Remark. The proof of the Margulis Lemma (cf. [3, §8.3]) is based on the fact that there are generators of N in a neighborhood of identity in which commutators are contracting with respect to some norm. It follows from this that for every δ > 0 there is K = K(δ, n) such that if γ is a k-fold commutator of these generators with k ≥ K then dγ(x) < δ. The following lemma is proved in [3, Lemma 7.4]. Lemma 2.2. Let N be a nilpotent group of isometries acting on a closed convex subset of X. Let Ns be the set of semisimple isometries ∩in N. Then Ns is a normal subgroup of N, and γ∈N Min(γ) is a non-empty N-invariant set. Lemma 2.3. Let 0 < ε < εn, and let N be the normal nilpotent subgroup of index ≤ mn in Γε(x). Then N is generated by elements in ∆2m ε(x). If Γε(x) is infinite then there is an element of infinite order nin N ∩ ∆2m ε(x). Furthermore, if Γε(x) contains parabolic elements nthen this element can be taken to be parabolic. Proof. Recall that if G is a group generated by a set of elements S, and H ≤ G is a subgroup of index r, then H is generated by words of length < 2r in the generators S. Indeed, if T = {t1, . . . , tr} is a transversal (i.e. a set of representatives of right cosets) of H in G, then H is generated by TST−1 ∩ H. Moreover, if T is a Schreier transversal (w.r.t. the generating set S) then its elements have length at most r−1 (for the last two statements see e.g. [13, §2.3]). Hence, H is generated by words of length 2(r − 1) + 1 < 2r. Let N be the normal nilpotent subgroup of Γε(x). Since [G : N] ≤ mn, N is generated by words of length < 2mn with respect to the generators of Γε(x). Hence it is generated by elements in ∆2m ε(x). nLet Ns be the set of semisimple elements in N. By 2.2, Ns is a sub- ∩<ul style="display: flex;"><li style="flex:1">group, and K = </li><li style="flex:1">Min(γ) is non-empty convex and N-invariant. </li></ul>γ∈Ns Notice that every elliptic element in N fixes K. Since N is infinite, it cannot fix K pointwise, and thus cannot be generated by elliptic elements. Thus, at least one element in N ∩ ∆2m ε(x) has infinite order. nIf Γε(x) contains a parabolic element, then so does N. In this case, at least one element in N ∩ ∆2m ε(x) is parabolic, otherwise N = Ns. n2.3. Elliptic isometries. For an elliptic isometry γ we write Fix(γ) for Min(γ). Let Γ be a group of isometries acting discretely on X. We ∩say Y is a singular submanifold if it equals γ∈A Fix(γ) for a set A ⊂ Γ of elliptic elements. We do not exclude the trivial case Y = X. <ul style="display: flex;"><li style="flex:1">BETTI NUMBERS OF FINITE VOLUME ORBIFOLDS </li><li style="flex:1">8</li></ul>Let Y, Y ′ be two totally geodesic submanifolds in X. If there exists γ ∈ Γ such that Y = γY ′ we say that Y, Y ′ are conjugate. Clearly, if Y is conjugate to a singular submanifold, then Y itself is a singular submanifold. nDefinition 2.4. An isometry g ∈ G = SO(n) of R is s-stable (s ∈ N) if CG(gi) = CG(g) (centralizer in G) and Fix(g) = Fix(gi) for i = 1, . . . , s. An elliptic isometry γ ∈ Isom(X) is s-stable if the isometry induced on the tangent bundle of a fixed point is s-stable. A singular submanifold Y is s-stable (w.r.t. to Γ) if there exists a subset A ⊂ Γ, such that every γ ∈ A is elliptic s-stable, and Y =

Betti Numbers of Finite Volume Orbifolds 10

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